Step-by-step explanation:
Take the first derivative
[tex] \frac{d}{dx} ( {x}^{3} - 3x)[/tex]
[tex]3 {x}^{2} - 3[/tex]
Set the derivative equal to 0.
[tex]3 {x}^{2} - 3 = 0[/tex]
[tex]3 {x}^{2} = 3[/tex]
[tex] {x}^{2} = 1[/tex]
[tex]x = 1[/tex]
or
[tex]x = - 1[/tex]
For any number less than -1, the derivative function will have a Positve number thus a Positve slope for f(x).
For any number, between -1 and 1, the derivative slope will have a negative , thus a negative slope.
Since we are going to Positve to negative slope, we have a local max at x=-1
Plug in -1 for x into the original function
[tex]( - 1) {}^{3} - 3( - 1) = 2 [/tex]
So the local max is 2 and occurs at x=-1,
For any number greater than 1, we have a Positve number for the derivative function we have a Positve slope.
Since we are going to decreasing to increasing, we have minimum at x=1,
Plug in 1 for x into original function
[tex]{1}^{3} - 3(1)[/tex]
[tex]1 - 3 = - 2[/tex]
So the local min occurs at -2, at x=1
